Given an array $a$, we have to find product of $a_{j}$-$a_{i}$ modulo $998244353$ over all $i$ and $j$ given $j>i$.
For eg. Let the array be $1,2,3$ then my answer will be calculated as-
As number of elements in the array could be large (upto $10^5$) I am looking for solution of order $nlogn$.
I have tried representing array as a polynomial but could get anything out of it. Please help.

  • $\begingroup$ To get an idea, start by computing (a+b+c)(a+b+c). $\endgroup$ – Hendrik Jan Jul 5 at 21:10
  • $\begingroup$ I don't think it will help as I want product of difference and not sum in the array. $\endgroup$ – Viplaw Srivastava Jul 5 at 21:25
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    $\begingroup$ Can you credit the original source where you encountered this task? $\endgroup$ – D.W. Jul 6 at 1:05
  • $\begingroup$ @ViplawSrivastava Oops, sorry, you are right. $\endgroup$ – Hendrik Jan Jul 6 at 10:58

The product $$V=\prod_{i<j}(a_j-a_i)$$ is the determinant of the Vandermonde matrix of the numbers $a_1,a_2,...,a_n$.

The square of this number is the discriminant $D$ of the polynomial $$p(x)=\prod_i(x-a_i)$$

This in turn is equal to $$V^2=D=(-1)^{n(n-1)/2}\prod_ip'(a_i)$$

You can quickly compute the coefficients of $p(x)$ and thus $p'(x)$, evaluate it at the $n$ points $a_1,a_2,...,a_n$ and their product. Then compute square root. The sign of $V$ you determine by sorting and counting the parity of the number of switches.

You can do all this in $O(n\log^2(n))$.

See the relevant algorithms here.

For the modular square root, you can use Tonelli-Shanks for efficiency. Although the theoretic order of this step is constant since the prime $998244353$ is fixed.

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  • $\begingroup$ @ViplawSrivastava FFT does that. A book that has the algorithms for all these steps is Aho, Hopcroft,Ullman The design and analysis of computer algorithms. For lack of a book, it seems that that particular step is mentioned here. $\endgroup$ – plop Jul 5 at 23:03

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